His mother married Sinatra, who was 29 years her senior, in Las Vegas in 1966. However he demanded a divorce two years later while she was filming Rosemary's Baby, the acclaimed thriller.

Mia Farrow with Frank Sinatra in the '60's (EPA)

Farrow later embarked on a 12-year relationship with Allen, while also starring in a string of his films, in the 1980s. However, the couple was never married and they lived in separate homes.

Any confirmation that he is not related to Allen is likely to be welcomed by Mr Farrow. The Yale University graduate and former youth adviser to Hillary Clinton at the State Department has repeatedly criticised his supposed father in recent years.

The Oscar-winning director separated from Mr Farrow's mother in 1992 after having an affair with Soon-Yi Previn, one of the actress's adopted daughters, who was then 22.

Farrow notoriously learned of the affair by discovering Allen possessed nude photographs of Ms Previn – a development the director described as "one of the great pieces of luck in my life".

The Farrows have never forgiven Allen, now 77, who has been married to Ms Previn, now 42, since 1997 and has two adopted daughters with her.

"Happy father's day – or as they call it in my family, happy brother-in-law's day," Mr Farrow said on Twitter in June last year. He has described Allen's actions as "a moral transgression".

Named Satchel Ronan O'Sullivan Farrow when he was born in 1987, Mr Farrow was presumed to be the sole biological child of Allen and Farrow, who were together from 1980.

The couple also adopted two children together. In all, Farrow has four biological children and nine adopted children. Two other adopted children have died.

The article, based on interviews with Farrow and several of her children, opened several old family wounds.

Dylan Farrow, Allen and Farrow's adopted daughter, used it to revive allegations that she was sexually molested by Allen as a young girl.

"I was seven," she said. "I was doing it because I was scared. I wanted it to stop." She assumed "this was how fathers treated their daughters," she said. "This was normal interaction, and I was not normal for feeling uncomfortable about it." Following an investigation by child services officials, no charges were brought against Allen. His lawyer said that he continued to deny the allegations.

Dylan Farrow added that Allen had attempted to contact her twice, but that she refused to have any contact with him or even to say his name. "I'm scared of him, his image," she said.

Fletcher Previn, another of Farrow's sons, said he had Photoshopped Allen out of every family photograph and edited him out of family videos, so that they would never have to see him again.

Meanwhile André Previn, Mia Farrow's husband between 1970 and 1979, said of Soon-Yi Previn, his adopted daughter and Allen's wife: "She does not exist."